# Spurious Vanishing Problem in Approximate Vanishing Ideal

arXiv: https://arxiv.org/abs/1901.08798 · HTML: https://mlsanity.com/paper/1901.08798/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jan 2019.

## Built on by (most cited first, 7 of 7)

- [Learning to Compute Gr"obner Bases](https://mlsanity.com/paper/2311.12904/) (2023)
- [Computational Algebra with Attention: Transformer Oracles for Border Basis Algorithms](https://mlsanity.com/paper/2505.23696/) (2025)
- [Approximate Vanishing Ideal Computations at Scale](https://mlsanity.com/paper/2207.01236/) (2022)
- [Border basis computation with gradient-weighted normalization](https://mlsanity.com/paper/2101.00401/) (2021)
- [Approximating Latent Manifolds in Neural Networks via Vanishing Ideals](https://mlsanity.com/paper/2502.15051/) (2025)
- [Conditional Gradients for the Approximate Vanishing Ideal](https://mlsanity.com/paper/2202.03349/) (2022)
- [Vanishing Component Analysis with Contrastive Normalization](https://mlsanity.com/paper/2210.16171/) (2022)

## Abstract (arXiv)

Approximate vanishing ideal is a concept from computer algebra that studies the algebraic varieties behind perturbed data points. To capture the nonlinear structure of perturbed points, the introduction of approximation to exact vanishing ideals plays a critical role. However, such an approximation also gives rise to a theoretical problem---the spurious vanishing problem---in the basis construction of approximate vanishing ideals; namely, obtained basis polynomials can be approximately vanishing simply because of the small coefficients. In this paper, we propose a first general method that enables various basis construction algorithms to overcome the spurious vanishing problem. In particular, we integrate coefficient normalization with polynomial-based basis constructions, which do not need the proper ordering of monomials to process for basis constructions. We further propose a method that takes advantage of the iterative nature of basis construction so that computationally costly operations for coefficient normalization can be circumvented. Moreover, a coefficient truncation method is proposed for further accelerations. From the experiments, it can be shown that the proposed method overcomes the spurious vanishing problem, resulting in shorter feature vectors while sustaining comparable or even lower classification error.